risks Article The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation Yuguang Fan 1,3, Philip S. Griffin 2, Ross Maller 3,*, Alexander Szimayer 4 and Tiandong Wang 5 1 ARC Centre of Excellence for Mathematical and Statistical Frontiers, School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia; [email protected] 2 Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, USA; psgriffi@syr.edu 3 Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, ACT 0200, Australia 4 School of Economics and Social Science, Universität Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany; [email protected] 5 School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA; [email protected] * Correspondence: [email protected] Academic Editor: Luca Regis Received: 21 November 2016; Accepted: 28 December 2016; Published: 6 January 2017 Abstract: We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer’s point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company’s ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this. Keywords: largest claims reinsurance; excess of loss reinsurance; ruin probability; ruin time; compound Poisson risk model; heavy tails; Lévy insurance risk process 1. Introduction The classical insurance risk model for a company employs a compound Poisson process with negative drift as the claims surplus process, and measures the lifetime of the company as the time taken for the value of the process to exceed the initial capital of the firm; the “ruin time”. Originally developed under a light tailed Cramér condition, in recent decades a wider spectrum of claim distributions—light, medium and heavy tailed—has been analysed, and, more generally, a Lévy process has been used in place of the compound Poisson process. Risks 2017, 5, 3; doi:10.3390/risks5010003 www.mdpi.com/journal/risks Risks 2017, 5, 3 2 of 27 A need for heavy tailed insurance risk models has been stressed, for example, by [1–3], and in this context, special interest lies in the possibility of reinsurance, whereby the company can hedge its risk of suffering extremely large claims. A reinsurance scheme increases its potential lifetime, thereby reducing the company’s risk of default. However, reinsurance treaties come at a cost, and pricing of those contracts and the consequent impact on the company’s overall value need to be considered. In this paper we investigate how reinsurance can extend the lifetime of the company and reduce the probability of ruin, with attention not just to heavy tailed claim distributions, but also to a variety of other possible distributional tail behaviours. Reinsurance works by transferring responsibility for some portion of the claims in a specified time period from the primary insurance company (the cedant) to the reinsurer. Two types of reinsurance which guard against the possibility of extremely large claims are excess of loss (EOL), and largest claim reinsurance (LCR). Each of these transfers the payment of part, or all, of one or more of the largest claims from the cedant to a reinsurer. A considerable amount of work has been done on these and related methods, usually taking the point of view of the reinsurer. Here, by contrast, we concentrate on the properties of the resulting reduced process from the point of view of the cedant and consider the relative merits of each type. To illustrate the effects, we analyse compound Poisson models for an insurance risk process incorporating an EOL or LCR aspect, or neither, computing ruin times and probabilities of ruin both in finite and infinite time scenarios. Using a dividend discounting model, we also determine the maximal amount the cedant is able to divert from dividend payments to the reinsurance premium, without reducing company value. To cover the spectrum of possibilities, as claims distributions we consider subexponential (including Pareto) distributions, as typifying heavy tailed situations, convolution equivalent distributions (such as the Inverse Gaussian) for medium, and distributions satisfying a Cramér condition (we use a Gamma distribution), for light tailed cases. In this way, much insight into the behaviour of the ruin time and associated quantities, such as the shortfall at ruin, can be gained. The paper is organised as follows. The EOL and LCR reinsurance models are reviewed in Section2. Section3 outlines our methods, with the compound Poisson model in Section 3.1, and the tail regimes we consider in Section 3.2. Section4 gives the results of the simulations, separately for LCR (Section 4.1) and EOL reinsurance (Section 4.2). Section 4.3 compares results across the distributions for both kinds of reinsurance. In Section5 we set out the dividend discounting model which is our basis for valuation of the cedant company, and use it to find the amount of the dividend the cedant is able to transfer to reinsurance without reducing the value of the company. This value is then simulated under the various regimes and conditions and comparisons made between the EOL and LCR strategies. Section6 contains a summary discussion of our results with suggestions for future research. In an Appendix we state some useful results concerning Laplace transforms of passage times which can be used to check on some aspects of the simulations, or provide bounds for quantities of interest. 2. Reinsurance Models A primary incentive for an insurance company to enter a reinsurance contract is to gain some degree of certainty over its cash flows. There are of course many ways in which risks can be transferred from cedant company to reinsurer. We briefly outline the two methods of reinsurance we will consider. Excess of Loss Reinsurance: Under this scheme, a retention amount L is pre-determined and the amount of any claim in excess of L is liable for the reinsurer. This scheme in effect truncates all claims at the level L, and the modified aggregate claims process is then simply the sum of the truncated claims. Analysis, both theoretical and practical, is relatively straightforward. A potential problem with this procedure, however, is the moral hazard it may give rise to. Moral hazard refers to changes in the cedant’s behaviour that may occur after having taken out Risks 2017, 5, 3 3 of 27 reinsurance; it may lead to less cautious behaviour and consequently to an increase in the potential magnitude and/or probability of a large loss. The work of [4,5], for example, discusses the issues involved in this, and how their effects may be disentangled empirically. Largest Claims Reinsurance: There are various alternatives to using a fixed retention level, usually based on making the insurer liable for a proportion of the total loss in some way. Here we examine the LCR treaty: having set a fixed follow-up time t, we delete from the process the largest claim occurring up to and including that time. Defined in this way, the scheme incorporates a retrospective feature akin to the construction of a “lookback” option as understood in finance 1. The reduced process constitutes a “trimmed” process, in which some part of, or all of, one or more of the largest claims has been deleted. Changes in the ruin probabilities and the expected ruin times of the cedant due to the trimming are then of particular interest. Ruin: “Ruin” occurs if the modified claim surplus process, starting from 0, exceeds the initial capital level u. The ruin time and consequent quantities are then calculated on the modified risk process. In Figure1, we provide graphical realisations of the LCR reinsurance scheme for one particular claim distribution, a Pareto(1, 2) (precise definitions of distributions are given in Section 3.2). The black points in Figure1 indicate individual claims arriving sequentially in time and the red segments represent the amounts that will be covered by reinsurance. 70 60 50 40 Claims 30 20 10 0 0 20 40 60 80 100 Time Pareto(1,2) Figure 1. A schematic illustration of the largest claim reinsurance (LCR) reinsurance scheme with a Pareto(1,2) claim distribution. Black dots indicate claim amounts and red lines are the successive amounts liable for the reinsurance company. Translating this scheme into the sample path of the cedant’s insurance risk process, we then have the illustration in Figure2, where the black line stands for the original risk process and the red line is the process adjusted for LCR reinsurance.